Investors (e.g. individuals, institutions, pension plans, mutual fund managers, and insurance companies) want to achieve high investment returns. However, many investors have failed to achieve this goal. From the collapse of Long-Term Capital Management in 1998, through the world financial crisis triggered by subprime lending, through current state pension funding crises, long-term performance has been elusive. See for example: http://www.nytimes.com/2013/03/12/business/sec-accuses-illinois-of-securities-fraud.html?_r=0&pagewanted=print
Over the last half-century or so, a number of mathematical approaches to portfolio construction have been developed and become well-known. The concept of a risk vs. return trade-off (exemplified in Harry Markowitz's Modem Portfolio Theory as the concept of “efficient frontier”) is perhaps the best example. The concept of a risk vs. return tradeoff is so pervasive in modem investment thinking that it has been adopted in investment law: see, for example, page 1 of the Uniform Prudent Investor Act at: http://www.uniformlaws.org/shared/docs/prudent%20investor/upia_final_94.pdf where it is stated “The tradeoff in all investing between risk and return is identified as the fiduciary's central consideration”.
Given a specific set of mathematical assumptions including assumed parameter values, there is a unique portfolio that on an expected-value basis outperforms every other portfolio in the long run, and is not systematically outperformed by other portfolios, even in the short run. This portfolio goes by many names in the literature, as described below, but hereinafter is referred to as “the optimal portfolio.” The assumed parameters include asset correlations and interest rates. In many applications, the assumed parameters are estimated from historical data or from market-implied values.
Implied values are parameter values that are inferred from available financial data rather than estimated from historical data. A good example is implied volatility, in which the volatility to use in the Black-Scholes option-pricing equation is “backed out” from observed option prices. Another good example is the implied correlation of forward interest rates, which can be backed out from observed swaption (option on an interest rate swap) prices. See: Options, Futures, and Other Derivative Securities by John Hull, Prentice Hall, © 1989, in which Section 5.9 deals with implied volatilities, and Volatility and Correlation in the Pricing of Equity, FX, and Interest-Rate Options by Riccardo Rebonato, Wiley, © 1999, in which Chapters 10 and 11 deal with implied volatilities and correlations in interest rate models.
The optimal portfolio satisfies the Kelly criterion (first published in 1956 as an analysis of optimal wagers in a favorable gambling game). See: A new interpretation of information rate, J. R. Kelly, Bell Syst. Techn. J. 35, 917-926, available at: http://www.bjmath.com/bjmath/kelly/kelly.pdf
The ideas are presented in a more traditional investment context in, for example, Chapters 9 and 15 of Investment Science (David Luenberger, Oxford University Press, (c) 1998).
The optimal portfolio is also described in: A Benchmark Approach to Investing and Pricing, by Eckhard Platen, available at: http://www.business.uts.edu.au/qfrc/research/research_papers/rp253.pdf
Key points regarding the optimal portfolio include:
1. The optimal portfolio is also known as the Growth-Optimal Portfolio, GOP, the Kelly portfolio, the log-optimal portfolio, or the numeraire portfolio.
2. In Platen's formulation, the optimal portfolio is strictly non-negative (no asset is shorted) and is constructed from the zeroth (risk-free) and risky assets numbered 1 to n. The non-negative proviso is not part of the standard definition of the optimal portfolio given by other researchers on the topic: we refer to this as the Platen restriction.
3. All nonnegative (long-only) portfolios denominated in terms of the Platen-restricted optimal portfolio (giving a “benchmarked value” of the portfolio) are supermartingales. “Supermartingale” is a term from probability theory meaning that on an expected value basis the benchmarked value of the portfolio (i.e. its value divided by the value of the optimal portfolio) stays the same or declines.
4. The optimal portfolio is optimal over specified intervals of time (i.e. the expected growth rate of all other nonnegative portfolios is no greater).
5. The optimal portfolio also has the highest long-term growth rate, i.e. the limiting behavior of the portfolio described in the point (4) above is as expected.
6. The optimal portfolio cannot be systematically outperformed (given a technical definition of “systematically outperformed”).
7. The market portfolio may differ from the optimal portfolio. This is a big departure from the usual thinking about the risk vs. return tradeoff, where any deviation from the market portfolio is taken as a “tilt” that increases risk.
8. There is a “real-world” asset price formula using the value of the optimal portfolio that does not depend on the existence of an equivalent risk-neutral probability measure. This is an important result. For example, option prices matching those obtained using the Black-Scholes formula can be obtained using a real-world measure rather than using the Black-Scholes risk-neutral assumptions. Measures are described in, for example Financial Calculus (Martin Baxter and Andrew Rennie, Cambridge University Press, © 1996).
9. For claims that do not depend on the risky assets, e.g. the price of a bond with a fixed coupon rate, there is a pricing formula based on zero-coupon bond prices. So despite the fact that the optimal portfolio is risky, simple claims (such as the value today of $1 due ten years from now) can be priced as usual.
10. Strong arbitrage (the existence of a portfolio that starts with zero capital and ends up with positive capital with a positive probability) is ruled out.
11. There is a Diversification Theorem, as follows: given enough assets and noise sources in the market, diversified portfolios approach the optimal portfolio. The formulation of the preceding theorem is complicated. The Naive Diversification Theorem given in Platen and Rendek's paper (see next paragraph) is easier to understand.
The Naive Diversification Theorem is given in Platen and Rendek's paper (Approximating the Numeraire Portfolio by Naive Diversification, Eckhard Platen and Renata Rendek, Research Paper 281 of the Quantitative Finance Research Centre at the University of Technology, Sydney, Australia). See: http://www.business.uts.edu.au/qfrc/research/research_papers/rp281.pdf
Hakansson and Ziemba, in section 3.1 of Capital Growth Theory (Nils H. Hakansson and William T. Ziemba, Chapter 3 in Handbooks in OR & MS, Vol. 9, © 1995 Elsevier Science B.V., available at http://www.hakansson.com/nils/papers/capita195.pdf) develop an additional key property of the optimal portfolio investment strategy: the strategy is “myopic”. By this the authors mean that only current-period returns and covariances are needed in order to achieve optimal behavior in the long run. This is to be distinguished from the dynamic programming approach commonly used to price options, for example, which would typically involve recursive, backwards-in-time solution of the portfolio composition and value given desired final conditions.
Luenberger (cited above) gives an explicit version of the two-fund theorem in the multi-period context. The theorem states that any efficient portfolio can be constructed from a mixture of the minimum variance portfolio and the optimal portfolio. In the case where there is a risk-free asset, Luenberger shows (equation 15.6) how to find the composition of the optimal portfolio if both long and short assets are permitted (i.e. ignoring the Platen restriction). Luenberger only deals with the case in which the asset return covariance matrix has full rank and is therefore invertible.
The optimal portfolio is also treated in A Benchmark Approach to Quantitative Finance (Eckhard Platen, David Heath, Springer, (c) 2006). Note that the authors' definition of the optimal portfolio as being necessarily nonnegative (referred to above as the Platen restriction) does not allow shorting of any asset and therefore differs from Luenberger's. The rationale for imposing the Platen restriction is that, in a jump diffusion model, any asset's value can jump to zero instantaneously (even if this happens very rarely), and if a short position were permitted the portfolio value could thereby become negative. Jump-diffusion models allow for market crashes and are therefore thought to be more realistic than those employing only geometric Brownian motion
A “leverage-constrained” optimal portfolio is a generalized definition encompassing both the Luenberger and Platen definitions of the optimal portfolio. A “leverage-constrained” optimal portfolio is one in which the amount of assets that can be shorted is greater than or equal to zero and less than or equal to the amount in the Luenberger solution. For example, this definition is applicable to the case in which an investor cannot borrow at the risk-free rate but must pay a spread over that rate because of institutional constraints or credit ratings.
As noted in above, the optimal portfolio has convenient numeraire properties for asset valuation. The word “numeraire” may require some explanation: it is defined in Financial Calculus (op. cit.) as “A basic security relative to which the value of other securities can be judged. Often the cash bond.” Here the numeraire is not the cash bond (or bank account), but a mixture of the risk-free and risky assets, possibly with a non-negativity constraint imposed.
It is well-known that stock options can be valued (using the Black-Scholes formula) as the expected discounted value of the option payoff under the so-called risk-neutral probability measure. Under this measure the stock index is assumed to grow at the risk-free rate (often taken to be the yield on United States (U.S.) government bonds or U.S. dollar denominated interest rate swaps—what Baxter and Rennie refer to as the “cash bond”) less the dividend rate. The risk-neutral measure is very explicitly not the real-world measure, although the two measures are linked by the Cameron-Martin-Girsanov transformation and the Radon-Nikodym derivative. See for example: http://www.chiark.greenend.org.uk/˜alanb/stoc-calc.pdf
Intuitively, the Radon-Nikodym derivative is the ratio of the risk-neutral probability density for the stock price to the real-world probability density for the stock price. In contrast it is possible to price options using the real-world measure if the optimal portfolio is used as numeraire (hence the term numeraire portfolio). This is a convenient alternative to risk-neutral pricing. Black-Scholes option prices can be reproduced using the real-world measure as shown below.
Given the existence of the optimal portfolio, the current benchmarked values of all assets are greater than or equal to their future benchmarked values: technically they are supermartingales as described above. All values are computed under the real-world, not risk-neutral, measure. This implies that no asset portfolio systematically outperforms the optimal portfolio. If the Platen restriction is in effect, then the correct statement is “no nonnegative asset portfolio systematically outperforms the optimal portfolio”.
If the Platen restriction is in effect, it can be shown (under suitable assumptions) that an equally-weighted world stock portfolio approaches a real-world proxy for the optimal portfolio in the limiting case. This is described in the Platen & Heath text cited above and in the Platen & Le paper Approximating the Growth Optimal Portfolio with a Diversified World Stock Index ((c) 2006), available at: http://www.business.uts.edu.au/qfrc/research/research_papers/rp184.pdf
Describing a portfolio in the limiting case and with known covariances and returns is of course an idealization. Real-world volatilities and returns change over time, and the limiting case of an infinite number of assets is never actually reached. This indicates that estimation and smoothing of returns and covariances, and efficient processing in the case of large numbers of assets, will be significant in any real-life application.
Another important point is that in the real world many investors are taxable (pension funds being a notable exception). Practical portfolio optimality may therefore, in addition to mathematical theory, require consideration of practical tax issues.
Some U.S. Tax Code considerations for international investment are outlined below.
A focus on international equity investment through a U.S.-domiciled mutual fund is appropriate for many U.S. investors. Although many investors concentrate their efforts on trying to identify and buy undervalued investments with the hope of selling them at a higher price, dividends can also contribute substantially to investment returns.
Not all dividends are taxed equally, however. The U.S. Tax Code distinguishes between qualified and non-qualified dividends. Typically, nonqualified dividends will be taxed at the taxpayer's marginal rate, which could be as high as 39.6% ignoring Medicare tax, whereas qualified dividends are taxed at 20%.
Dividends from publicly-traded U.S. corporations are usually qualified dividends. Dividends from publicly-traded corporations engaged in an active trade or business domiciled in countries with which the U.S. has a tax treaty are also usually qualified dividends.
U.S.-domiciled mutual funds investing in international assets are subject to more complex considerations. Foreign-domiciled mutual funds are not considered in this disclosure since such funds generally have a number of disadvantages under U.S. tax law, and Americans are generally prevented from investing in them under U.S. securities law.
The foreign jurisdictions with which the U.S. has tax treaties generally have withholding tax in place on dividend income, often at a rate of 15%. There are two main alternatives for fund shareholders to get credit for these taxes on their U.S. tax returns, reducing or even eliminating double taxation:
1. The foreign tax deduction: the fund shareholder can choose to deduct foreign taxes paid as an expense. So, for example, if the shareholder pays tax at a 40% marginal rate, the incremental tax reduction will be 40% of the foreign tax.
2. The foreign tax credit: the fund shareholder can credit foreign taxes against U.S. tax payable (within limits). This is clearly more favorable tax treatment for the fund shareholder than the foreign tax deduction.
According to the Bogleheads website at: http://www.bogleheads.org/wiki/Foreign_tax_credit, U.S. taxpayers cannot claim foreign tax credits for foreign mutual funds held in IRA's, 401(k)'s, or variable annuities (VA's). However the Allianz High Five variable annuity (VA) prospectus, available at: https://www.allianzlife.com/content/public/Literature/Documents/HFV-001.pdf says: “We may benefit from any foreign tax credits attributable to taxes paid by certain funds to foreign jurisdictions to the extent permitted under the federal tax law.”
The implication is clear: a life insurance carrier can benefit from the foreign tax credit for funds held in a separate account backing an ordinary VA.
Some of the U.S. tax code considerations noted above, as applied to a VA, may depend on the details of the separate account structure. The Bogleheads article says there is no credit for foreign taxes in a fund-of-funds structure, but that may not carry over to tax treatment for a carrier holding assets in separate account.
There are different U.S. tax code considerations for a tax flow-through annuity (see U.S. Pat. No. 7,716,075). The product is not tax-deferred, the carrier has legal title to the fund assets, and the key tax flow-through feature is that the annuity owner owns the funds for tax purposes (this is how capital gains treatment for sale of fund shares is achieved, for example). It therefore seems very likely that the tax flow-through annuity owner can claim the foreign tax credit.
If a diversified world fund's holdings are about 50% outside the U.S., with dividends being paid at 3% or so (but possibly much higher if the stocks held are for instance
Australian or Canadian), and 15% withholding applies (the standard tax treaty rate described above) then there will be a 0.225% per year pricing advantage for a variable annuity (VA) with such holdings. Although this may seem like a small advantage, over a 30-year holding period it could easily lead to account balances 6-7% higher than otherwise: for example, (1.0522530)÷1.0530=1.0663.
Although turning qualified dividends into ordinary income may be undesirable, putting high-dividend foreign stocks into a VA creates a product suited to a low domestic yield environment.
If one can select an international (that is, ex-U.S.) stock portfolio with a dividend yield of 4% or so, then credit for an assumed 15% withholding tax generates a tax credit of about 0.60%. This would work well for high-yielding Canadian stocks such as Telus or Bell-Alliant, for example.
If the carrier prices in full knowledge of the above tax credit, then other product expenses such as the mortality and expense risk charge (M&E), or more generally the fund management fee if multiple foreign funds are offered, can be reduced, providing higher tax-deferred yield to clients.